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- 1. ITFA-2007-24 The Geometer’s Toolkit to String CompactiﬁcationsarXiv:0706.1310v1 [hep-th] 9 Jun 2007 based on lectures given at the Workshop on String and M–Theory Approaches to Particle Physics and Astronomy Galileo Galilei Institute for Theoretical Physics Arcetri (Firenze) Susanne Reffert ITFA Amsterdam May 2007
- 2. Contents1 Calabi–Yau basics and orbifolds 3 1.1 Calabi–Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Orbifolds – A simple and common example . . . . . . . . . . . . . . . . . . . 6 1.2.1 Point groups and Coxeter elements . . . . . . . . . . . . . . . . . . . . 6 1.2.2 List of point groups and lattices . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Fixed set conﬁgurations and conjugacy classes . . . . . . . . . . . . . . 10 1.3 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Toric Geometry 14 2.1 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Resolution of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Mori cone and intersection numbers . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Divisor topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Application: Desingularizing toroidal orbifolds 28 3.1 Gluing the patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 The inherited divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 The intersection ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Divisor topologies for the compact manifold . . . . . . . . . . . . . . . . . . . 394 The orientifold quotient 44 4.1 Yet another quotient: The orientifold . . . . . . . . . . . . . . . . . . . . . . . 44 (1,1) 4.2 When the patches are not invariant: h− = 0 . . . . . . . . . . . . . . . . . . 45 4.3 The local orientifold involution . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 The intersection ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1
- 3. SummaryThe working string theorist is often confronted with the need to make use of various tech-niques of algebraic geometry. Unfortunately, a problem of language exists. The specializedmathematical literature is often difﬁcult to read for the physicist, moreover differences interminology exist. These lectures are meant to serve as an introduction to some geometric constructions andtechniques (in particular the ones of toric geometry) often employed by the physicist workingon string theory compactiﬁcations. The emphasis is wholly on the geometry side, not on thephysics. Knowledge of the basic concepts of differential, complex and Kähler geometry isassumed. The lectures are divided into four parts. Lecture one brieﬂy reviews the basics of Calabi–Yau geometry and then introduces toroidal orbifolds, which enjoy a lot of popularity in stringmodel building constructions. In lecture two, the techniques of toric geometry are introduced,which are of vital importance for a large number of Calabi–Yau constructions. In particular,it is shown how to resolve orbifold singularities, how to calculate the intersection numbersand how to determine divisor topologies. In lectures three, the above techniques are used toconstruct a smooth Calabi–Yau manifold from toroidal orbifolds by resolving the singularitieslocally and gluing together the smooth patches. The full intersection ring and the divisortopologies are determined by a combination of knowledge about the global structure of T 6 /Γand toric techniques. In lecture four, the orientifold quotient of such a resolved toroidalorbifold is discussed. The theoretical discussion of each technique is followed by a simple, explicit example.At the end of each lecture, I give some useful references, with emphasis on text books andreview articles, not on the original articles. 2
- 4. Lecture 1Calabi–Yau basics and orbifoldsContents 1.1 Calabi–Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Orbifolds – A simple and common example . . . . . . . . . . . . . . . . . 6 1.2.1 Point groups and Coxeter elements . . . . . . . . . . . . . . . . . . . 6 1.2.2 List of point groups and lattices . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Fixed set conﬁgurations and conjugacy classes . . . . . . . . . . . . . 10 1.3 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 In this lecture, I will brieﬂy review the basics of Calabi–Yau geometry. As a simple andextremely common example, I will introduce toroidal orbifolds.1.1 Calabi–Yau manifoldsCalabi conjectured in 1957 that a compact Kähler manifold X of vanishing ﬁrst Chern classalways admits a Ricci–ﬂat metric. This was proven by Yau in 1977. Such a manifold X ofdimension n is now known as Calabi–Yau manifold. Equivalently, X is Calabi–Yau if it (a) admits a Levi–Civita connection with SU (n) holonomy (b) admits a nowhere vanishing holomorphic (n, 0)–form Ω (c) has a trivial canonical bundle. The Hodge numbers of a complex manifold are often displayed in a so–called Hodge dia-mond: h0,0 h 1,0 h0,1 h 2,0 h 1,1 h0,2 h3,0 h2,1 h1,2 h0,3 (1.1) h3,1 h2,2 h1,3 h3,2 h2,3 h3,3 3
- 5. For a Kähler manifold, the Hodge diamond has two symmetries: • Complex conjugation ⇒ h p,q = hq,p (vertical reﬂection symmetry), • Poincaré duality ⇒ h p,q = hn−q,n− p (horizontal reﬂection symmetry).For X being Calabi–Yau, the Hodge diamond is even more constrained: (b) implies thathn,0 = 1 and furthermore h p,0 = hn− p,0 . The Hodge–diamond of a Calabi–Yau 3–fold thereforetakes the form 1 0 0 0 h1,1 0 1 h2,1 h2,1 1 (1.2) 0 h 1,1 0 0 0 1Thus, the Hodge numbers of X are completely speciﬁed by h1,1 and h2,1 . The Euler number ofX is χ( X ) = 2 (h1,1 ( X ) − h2,1 ( X )). (1.3)Until fairly recently, not a single example of an explicit compact Calabi–Yau metric wasknown! 1 A Calabi–Yau manifold can be deformed in two ways: Either by varying its complex struc-ture (its "shape"), or by varying its Kähler structure (its "size"). Variations of the metric ofmixed type δgmn correspond to variations of the Kähler structure and give rise to h1,1 param-eters, whereas variations of pure type δgmn , δgmn correspond to variations of the complexstructure and give rise to h2,1 complex parameters. To metric variations of mixed type, a real(1, 1)–form can be associated: i δgmn dzm ∧ dzn . (1.4)To pure type metric variations, a complex (2, 1)–form can be associated: Ωijk gkn δgmn dzi ∧ dz j ∧ dzm , (1.5)where Ω is the Calabi–Yau (3, 0)–form.1D Calabi–YausIt is easy to list all one–dimensional Calabi–Yaus: there is but the complex plane, the punc-tured complex plane (i.e.the cylinder) and the two–torus T 2 . The Hodge diamond of a 1D Calabi–Yau is (not surprisingly) completely constrained: h0,0 1 h1,0 h0,1 = 1 1 (1.6) h1,1 1 1 This only changed with the introduction of Calabi–Yaus that are cones over Sasaki–Einstein manifolds, seee.g. [1]. 4
- 6. We now illustrate the concept of moduli for the simple case of T 2 , which has the metric g11 g12 R21 R1 R2 cos θ12 g= = . (1.7) g12 g22 R1 R2 cos θ12 R22A T 2 comes with one Kähler modulus T , which parametrizes its volume, and one complexstructure modulus, which corresponds to its modular parameter U = τ. Figure 1.1 depicts Im(z) R2/R1sinθ τ=R2/R1eiθ Re(z) 1 Figure 1.1: Fundamental region of a T 2the fundamental region of a T 2 . The area of the torus is given by R1 R2 sin θ12 , expressedthrough the metric, we ﬁnd T = det g = R1 R2 sin θ12 . (1.8)In heterotic string theory, the Kähler moduli are complexiﬁed by pairing them up with thecomponents of the anti–symmetric tensor B. In type I IB string theory, the Kähler moduli arepaired with the components of the Ramond–Ramond four–form C4 . The usual normalizationof the fundamental region in string theory is such that the a–cycle is normalized to 1, whilethe modular parameter becomes τ = R2 /R1 eiθ . The complex structure modulus expressedthrough the metric is 1 U= ( g12 + i det g ). (1.9) g112D Calabi–YausIn two dimensions, there are (up to diffeomorphism) only two compact Calabi–Yaus: theK3–surface and the 4–torus T 4 . The Hodge diamond of the K3 is h0,0 1 h1,0 h0,1 0 0 h 2,0 h1,1 h0,2 = (1.10) 1 20 1 h2,1 h1,2 0 0 h2,2 1 5
- 7. 3D Calabi–YausIn three dimensions, no classiﬁcation exists. It is not even known whether there are ﬁnitelyor inﬁnitely many (up to diffeomorphism). There are several classes, which can be constructed fairly easily: • hypersurfaces in toric varieties • complete intersections in toric varieties (CICY) • toroidal orbifolds and their resolutions • Cones over Sasaki–Einstein spaces (with metric!). In the following, I will mainly concentrate on the third point. The machinery of toricgeometry will be introduced, which is vital for most of these constructions.1.2 Orbifolds – A simple and common exampleThe string theorist has been interested in orbifolds for many years already (see [2] in 1985),and for varying reasons, one of course being their simplicity. Knowledge of this constructionis thus one of the basic requirements for a string theorist. An orbifold is obtained by dividing a smooth manifold by the non–free action of a discretegroup: X = Y/Γ . (1.11)The original mathematical deﬁnition is broader: any algebraic variety whose only singulari-ties are locally of the form of quotient singularities is taken to be an orbifold. The string theorist is mostly concerned with toroidal orbifolds of the form T 6 /Γ. While thetorus is completely ﬂat, the orbifold is ﬂat almost everywhere: its curvature is concentratedin the ﬁxed points of Γ. At these points, conical singularities appear. Only the simplest variety of toroidal orbifolds will be discussed here: Γ is taken to beabelian, there will be no discrete torsion or vector structure. Toroidal orbifolds are simple, yet non–trivial. Their main asset is calculability, which holdsfor purely geometric as well as for string theoretic aspects.1.2.1 Point groups and Coxeter elementsA torus is speciﬁed by its underlying lattice Λ: Points which differ by a lattice vector areidentiﬁed: x ∼ x + l, l ∈ Λ . (1.12)The six–torus is therefore deﬁned as quotient of R6 with respect to the lattice Λ: T 6 = R6 /Λ . (1.13)To deﬁne an orbifold of the torus, we divide by a discrete group Γ, which is called the pointgroup, or simply the orbifold group. We cannot choose any random group as the point groupΓ, it must be an automorphism of the torus lattice Λ, i.e. it must preserve the scalar productand fulﬁll g l ∈ Λ if l ∈ Λ, g ∈ Γ . (1.14) 6
- 8. To fully specify a toroidal orbifold, one must therefore specify both the torus lattice as wellas the point group. In the context of string theory, a set–up with SU (3)–holonomy2 is whatis usually called for, which restricts the point group Γ to be a subgroup of SU (3). Since werestrict ourselves to abelian point groups, Γ must belong to the Cartan subalgebra of SO(6).On the complex coordinates of the torus, the orbifold twist will act as θ : (z1 , z2 , z3 ) → (e2πiζ 1 z1 , e2πiζ 2 z2 , e2πiζ 3 z3 ), 0 ≤ |ζ i | < 1, i = 1, 2, 3. (1.15)The requirement of SU (3)–holonomy can also be phrased as requiring invariance of the(3, 0)–form of the torus, Ω = dz1 ∧ dz2 ∧ dz3 . This leads to ± ζ 1 ± ζ 2 ± ζ 3 = 0. (1.16)We must furthermore require that Γ acts crystallographically on the torus lattice. Togetherwith the condition (1.16), this amounts to Γ being either Z N with N = 3, 4, 6, 7, 8, 12 , (1.17)or Z N × Z M with M a multiple of N and N = 2, 3, 4, 6. With the above, one is lead to theusual standard embeddings of the orbifold twists, which are given in Tables 1.1 and 1.2. Themost convenient notation is 1 (ζ 1 , ζ 2 , ζ 3 ) = (n1 , n2 , n3 ) with n1 + n2 + n3 = 0 mod n . (1.18) nNotice that Z6 , Z8 and Z12 have two inequivalent embeddings in SO(6). 1 Point group n ( n1 , n2 , n3 ) Z3 1 3 (1, 1, −2) Z4 1 4 (1, 1, −2) Z6 − I 1 6 (1, 1, −2) Z6 − I I 1 6 (1, 2, −3) Z7 1 7 (1, 2, −3) Z8 − I 1 8 (1, 2, −3) Z8 − I I 1 8 (1, 3, −4) Z12− I 1 12 (1, 4, −5) Z12− I I 1 12 (1, 5, −6) Table 1.1: Group generators for Z N -orbifolds. For all point groups given in Tables 1.1 and 1.2, it is possible to ﬁnd a compatible toruslattice, in several cases even more than one. We will now repeat the same construction starting out from a real six–dimensional lattice.A lattice is suitable for our purpose if its automorphism group contains subgroups in SU (3). 2 Thisresults in N = 1 supersymmetry for heterotic string theory and in N = 2 in type I I string theories infour dimensions. 7
- 9. 1 1 Point group n ( n1 , n2 , n3 ) m ( m1 , m2 , m3 ) Z2 × Z2 1 2 (1, 0, −1) 1 2 (0, 1, −1) Z2 × Z4 1 2 (1, 0, −1) 1 4 (0, 1, −1) Z2 × Z6 1 2 (1, 0, −1) 1 6 (0, 1, −1) Z2 × Z6 1 2 (1, 0, −1) 1 6 (1, 1, −2) Z3 × Z3 1 3 (1, 0, −1) 1 3 (0, 1, −1) Z3 × Z6 1 3 (1, 0, −1) 1 6 (0, 1, −1) Z4 × Z4 1 4 (1, 0, −1) 1 4 (0, 1, −1) Z6 × Z6 1 6 (1, 0, −1) 1 6 (0, 1, −1) Table 1.2: Group generators for Z N × Z M -orbifolds.Taking the eigenvalues of the resulting twist, we are led back to twists of the form (1.17). Apossible choice is to consider the root lattices of semi–simple Lie–Algebras of rank 6. All oneneeds to know about such a lattice is contained in the Cartan matrix of the respective Liealgebra. The matrix elements of the Cartan matrix are deﬁned as follows: ei , e j Aij = 2 , (1.19) ej, ejwhere the ei are the simple roots. The inner automorphisms of these root lattices are given by the Weyl–group of the Lie–algebra. A Weyl reﬂection is a reﬂection on the hyperplane perpendicular to a given root: x, ei Si ( x ) = x − 2 ei . (1.20) ei , eiThese reﬂections are not in SU (3) and therefore not suitable candidates for a point group,but the Weyl group does have a subgroup contained in SU (3): the cyclic subgroup generatedby the Coxeter element, which is given by successive Weyl reﬂections with respect to all simpleroots: Q = S1 S2 ...Srank . (1.21)The so–called outer automorphisms are those which are generated by transpositions of rootswhich are symmetries of the Dynkin diagram. By combining Weyl reﬂections with such outerautomorphisms, we arrive at so–called generalized Coxeter elements. Pij denotes the transpo-sition of the i’th and j’th roots. The orbifold twist Γ may be represented by a matrix Qij , which rotates the six lattice basisvectors:3 ei → Q ji e j . (1.22)The following discussion is restricted to cases in which the orbifold twist acts as the (general-ized) Coxeter element of the group lattices, these are the so–called Coxeter–orbifolds4 . 3 Different symbols for the orbifold twist are used according to whether we look at the quantity which acts onthe real six-dimensional lattice ( Q) or on the complex coordinates (θ ). 4 It is also possible to construct non–Coxeter orbifolds, such as e.g. Z on SO (4)3 as discussed in [3]. 4 8
- 10. We now change back to the complex basis {zi }i=1,2,3 , where the twist Q acts diagonallyon the complex coordinates, i.e. θ : zi → e2πiζ i zi , (1.23)with the eigenvalues 2πi ζ i introduced above. To ﬁnd these complex coordinates we makethe ansatz z i = a1 x 1 + a2 x 2 + a3 x 3 + a4 x 4 + a5 x 5 + a6 x 6 . i i i i i i (1.24)Knowing how the Coxeter twist acts on the root lattice and therefore on the real coordinatesxi , and knowing how the orbifold twist acts on the complex coordinates, see Tables 1.3 and1.4, we can determine the coefﬁcients aij by solving Qt zi = e2πiζ i zi . (1.25)The transformation which takes us from the real to the complex basis must be unimodular.The above equation only constrains the coefﬁcients up to an overall complex normalizationfactor. For convenience we choose a normalization such that the ﬁrst term is real.Example A: Z6− I on G2 × SU (3) 2 2We take the torus lattice to be the root lattice of G2 × SU (3), a direct product of three ranktwo root lattices, and explicitly construct its Coxeter element. First, we look at the SU (3)–factor.With the Cartan matrix of SU (3), 2 −1 A= , (1.26) −1 2and eq. (1.20), the matrices of the two Weyl reﬂections can be constructed: −1 1 1 0 S1 = , S2 = . (1.27) 0 1 1 −1The Coxeter element is obtained by multiplying the two: 0 −1 QSU (3) = S1 S2 = . (1.28) 1 −1In the same way, we arrive at the Coxeter-element of G2 . The six-dimensional Coxeter element isbuilt out of the three 2 × 2–blocks: 2 −1 0 0 0 0 3 −1 0 0 0 0 0 0 2 −1 0 0 Q= . (1.29) 0 0 3 −1 0 0 0 0 0 0 0 −1 0 0 0 0 1 −1The eigenvalues of Q are e2πi/6 , e−2πi/6 , e2πi/6 , e−2πi/6 , e2πi/3 , e−2πi/3 , i.e. those of the Z6− I –twist, see Table 1.1, and Q fulﬁlls Q6 = Id. Solving (1.25) yields the following solution for the complex coordinates: z1 = a (−(1 + e2πi/6 ) x1 + x2 ) + b (−(1 + e2πi/6 ) x3 + x4 ), 9
- 11. z2 = c (−(1 + e2πi/6 ) x1 + x2 ) + d (−(1 + e2πi/6 ) x3 + x4 ), z3 = e (e2πi/3 x5 + x6 ), (1.30)where a, b, c, d and e are complex constants left unﬁxed by the twist alone. In the following, wewill choose a, d, e such that x1 , x3 , x5 have a real coefﬁcient and the transformation matrix isunimodular and set b = c = 0, so the complex structure takes the following form: 1 z1 = x1 + √ e5πi/6 x2 , 3 1 z2 = x3 + √ e5πi/6 x4 , 3 z = 3 ( x + e2πi/3 x6 ). 3 1/4 5 (1.31)1.2.2 List of point groups and latticesIn the Tables 1.3 and 1.4, a list of torus lattices together with the compatible orbifold pointgroup is given [4].5 Notice that some point groups are compatible with several lattices. The tables give the torus lattices and the twisted and untwisted Hodge numbers. Thelattices marked with , , and ∗ are realized as generalized Coxeter twists, the automorphismbeing in the ﬁrst and second case S1 S2 S3 S4 P36 P45 and in the third S1 S2 S3 P16 P25 P34 .1.2.3 Fixed set conﬁgurations and conjugacy classesMany of the deﬁning properties of an orbifold are encoded in its singularities. Not only thetype (which group element they come from, whether they are isolated or not) and number ofsingularities is important, but also their spatial conﬁguration. Here, it makes a big differenceon which torus lattice a speciﬁc twist lives. The difference does not arise for the ﬁxed pointsin the ﬁrst twisted sector, i.e. those of the θ–element which generates the group itself. But inthe higher twisted sectors, in particular in those which give rise to ﬁxed tori, the number ofﬁxed sets differs for different lattices, which leads to differing Hodge numbers. A point f (n) is ﬁxed under θ n ∈ Zm , n = 0, ..., m − 1, if it fulﬁlls θ n f (n) = f (n) + l, l ∈ Λ, (1.32)where l is a vector of the torus lattice. In the real lattice basis, we have the identiﬁcation xi ∼ xi + 1 . (1.33)Like this, we obtain the sets that are ﬁxed under the respective element of the orbifold group. 1 1A twist n (n1 , n2 , n3 ) and its anti–twist n (1 − n1 , 1 − n2 , 1 − n3 ) give rise to the same ﬁxed sets,so do permutations of (n1 , n2 , n3 ). Therefore not all group elements of the point group needto be considered separately. The prime orbifolds, i.e. Z3 and Z7 have an especially simpleﬁxed point conﬁguration since all twisted sectors correspond to the same twist and so giverise to the same set of ﬁxed points. Point groups containing subgroups generated by elementsof the form 1 (n1 , 0, n2 ), n1 + n2 = 0 mod n (1.34) n 5 Other references such as [5] give other lattices as well. 10
- 12. ZN Lattice huntw. (1,1) huntw. (2,1) htwist. (1,1) htwist. (2,1) Z3 SU (3)3 9 0 27 0 Z4 SU (4)2 5 1 20 0 Z4 SU (2) × SU (4) × SO(5) 5 1 22 2 Z4 SU (2)2 × SO(5)2 5 1 26 6 Z6− I ( G2 × SU (3)2 ) 5 0 20 1 Z6− I 2 SU (3) × G2 5 0 24 5 Z6− I I SU (2) × SU (6) 3 1 22 0 Z6− I I SU (3) × SO(8) 3 1 26 4 Z6− I I (SU (2)2 × SU (3) × SU (3)) 3 1 28 6 Z6− I I SU (2)2 × SU (3) × G2 3 1 32 10 Z7 SU (7) 3 0 21 0 Z8− I (SU (4) × SU (4))∗ 3 0 21 0 Z8− I SO(5) × SO(9) 3 0 24 3 Z8− I I SU (2) × SO(10) 3 1 24 2 Z8− I I SO(4) × SO(9) 3 1 28 6 Z12− I E6 3 0 22 1 Z12− I SU (3) × F4 3 0 26 5 Z12− I I SO(4) × F4 3 1 28 6 Table 1.3: Twists, lattices and Hodge numbers for Z N orbifolds. ZN Lattice huntw. (1,1) huntw. (2,1) htwist. (1,1) htwist. (2,1) Z2 × Z2 SU (2)6 3 3 48 0 Z2 × Z4 SU (2)2 × SO(5)2 3 1 58 0 Z2 × Z6 SU (2)2 × SU (3) × G2 3 1 48 2 Z2 × Z6 SU (3) × G22 3 0 33 0 Z3 × Z3 SU (3) 3 3 0 81 0 Z3 × Z6 SU (3) × G22 3 0 70 1 Z4 × Z4 SO(5) 3 3 0 87 0 Z6 × Z6 G23 3 0 81 0Table 1.4: Twists, lattices and Hodge numbers for Z N × Z M orbifolds. 11
- 13. give rise to ﬁxed tori. It is important to bear in mind that the ﬁxed points were determined on the coveringspace. On the quotient, points which form an orbit under the orbifold group are identiﬁed.For this reason, not the individual ﬁxed sets, but their conjugacy classes must be counted. To form a notion of what the orbifold looks like, it is useful to have a schematic picture ofthe conﬁguration, i.e. the intersection pattern of the singularities.Example A: Z6− I on G2 × SU (3) 2In the following, we will identify the ﬁxed sets under the θ–, θ 2 – and θ 3 –elements. θ 4 and θ 5yield no new information, since they are simply the anti–twists of θ 2 and θ. The Z6− I –twist hasonly one ﬁxed point in each torus, namely zi = 0. The Z3 –twist has three ﬁxed points in each √ √direction, namely z1 = z2 = 0, 1/3, 2/3 and z3 = 0, 1/ 3 eπi/6 , 1 + i/ 3. The Z2 –twist, which 1arises in the θ 3 –twisted sector, has four ﬁxed points, corresponding to z1 = z2 = 0, 2 , 1 τ, 1 (1 + τ ) 2 2for the respective modular parameter τ. As a general rule, we shall use red to denote the ﬁxedset under θ, blue to denote the ﬁxed set under θ 2 and pink to denote the ﬁxed set under θ 3 . Notethat the ﬁgure shows the covering space, not the quotient. Table 1.5 summarizes the important data of the ﬁxed sets. The invariant subtorus under θ 3is (0, 0, 0, 0, x5 , x6 ) which corresponds simply to z3 being invariant. Group el. Order Fixed Set Conj. Classes θ = 1 (1, 1, 4) 6 6 3 ﬁxed points 3 1 θ2 = 3 (1, 1, 1) 3 27 ﬁxed points 15 1 θ3 = 2 (1, 1, 0) 2 16 ﬁxed lines 6 Table 1.5: Fixed point set for Z6− I on G2 × SU (3) 2 Figure 1.2: Schematic picture of the ﬁxed set conﬁguration of Z6− I on G2 × SU (3) 2 12
- 14. Figure 1.2 shows the conﬁguration of the ﬁxed sets in a schematic way, where each complexcoordinate is shown as a coordinate axis and the opposite faces of the resulting cube of length 1are identiﬁed. Note that this ﬁgure shows the whole six–torus and not the quotient. The arrowsindicate the orbits of the ﬁxed sets under the action of the orbifold group.1.3 LiteratureA very good introduction to complex manifolds are the lecture notes by Candelas and de laOssa [6], which are unfortunately not available online. Usually, old paper copies which werehanded down from earlier generations can still be found in most string theory groups. Fornearly all purposes, the book of Nakahara [7] is an excellent reference. An introduction to allnecessary basics which is very readable for the physicist is given in Part 1 of [8]. Speciﬁcallyfor Calabi–Yau geometry, there is the book by Hübsch [9]. A number of lecture notes andreviews contain much of the basics, see for example [10]. On Orbifolds, a number of reviews exist (mainly focusing on physics, though), e.g. [5].More orbifold examples as introduced above are contained in [11].Bibliography [1] J. P. Gauntlett, D. Martelli, J. Sparks, and D. Waldram, Sasaki-Einstein metrics on S(2) x S(3), Adv. Theor. Math. Phys. 8 (2004) 711–734, [hep-th/0403002]. [2] L. J. Dixon, J. A. Harvey, C. Vafa, and E. Witten, Strings on orbifolds, Nucl. Phys. B261 (1985) 678–686. [3] J. A. Casas, F. Gomez, and C. Munoz, Complete structure of Z(n) Yukawa couplings, Int. J. Mod. Phys. A8 (1993) 455–506, [hep-th/9110060]. [4] J. Erler and A. Klemm, Comment on the generation number in orbifold compactiﬁcations, Commun. Math. Phys. 153 (1993) 579–604, [hep-th/9207111]. [5] D. Bailin and A. Love, Orbifold compactiﬁcations of string theory, Phys. Rept. 315 (1999) 285–408. [6] P. Candelas and X. de la Ossa, Lectures on complex manifolds, . [7] M. Nakahara, Geometry, Topology and Physics. Graduate Student Series in Physics. Institute of Physics Publishing, Bristol and Philadelphia, 1990. [8] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry. No. 1 in Clay Mathematics Monographs. American Mathematical Society, Clay Mathematics Institute, 2003. [9] T. Hübsch, Calabi-Yau manifolds. A bestiary for physicists. World Scientiﬁc, Singapore, 1991.[10] B. R. Greene, String theory on Calabi-Yau manifolds, hep-th/9702155.[11] S. Reffert, Toroidal orbifolds: Resolutions, orientifolds and applications in string phenomenology, hep-th/0609040. 13
- 15. Lecture 2Toric GeometryContents 2.1 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Resolution of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Mori cone and intersection numbers . . . . . . . . . . . . . . . . . . . . . 21 2.4 Divisor topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 In this lecture, I will introduce an extremely useful tool, namely the methods of toricgeometry. The geometry is summarized in combinatorial data, which is fairly simple to use. After introducing the basics, I will discuss the resolution of singularities via a bow–up,the determination of the Mori generators and the intersection ring, as well as how to deter-mine the divisor topologies in non–compact toric varieties. The material is introduced at theexample of orbifolds of the form C3 /Zn .2.1 The basicsAn n–dimensional toric variety has the form XΣ = (C N FΣ )/(C∗ )m , (2.1)where m < N, n = N − m. (C∗ )m is the algebraic torus which lends the variety its name andacts via coordinatewise multiplication1 . FΣ is the subset that remains ﬁxed under a continuoussubgroup of (C∗ )m and must be subtracted for the variety to be well–deﬁned. Toric varieties can also be described in terms of gauged linear sigma models. In short, foran appropriate choice of Fayet–Iliopoulos parameters, the space of supersymmetric groundstates of the gauged linear sigma models is a toric variety. We will not take this point of viewhere and thus refer the reader to the literature, e.g. [3]. 1 An algebraic torus can be deﬁned for any ﬁeld K. The name is connected to the fact that if K = C, analgebraic torus is the complexiﬁcation of the standard torus (S1 )n . 14
- 16. Example 0: Projective spacesThe complex projective space Pn (sometimes also denoted CPn ) is deﬁned by Pn = (Cn+1 {0})/C∗ . (2.2)It is a quotient space and corresponds to the complex lines passing through the origin of Cn+1 .C∗ acts by coordinatewise multiplication. 0 has to be removed, so C∗ acts freely (without ﬁxedpoints). Pn thus corresponds to the space of C∗ orbits. Points on the same line are equivalent: [ X0 , X1 , ..., Xn ] ∼ [λ X0 , λ X1 , ..., λ Xn ], λ ∈ C∗ . (2.3)The X0 , ...Xn are the so–called homogeneous coordinates and are redundant by one. In a localcoordinate patch with Xi = 0, one can deﬁne coordinates invariant under rescaling zk = Xk /Xi , k = i. (2.4)Pn is compact and all its complex submanifolds are compact. Moreover, Chow proved that anysubmanifold of Pn can be realized as the zero locus of ﬁnitely many homogeneous polynomialequations. P1 corresponds to S2 . Weighted projective spaces are a generalization of the above, with different torus actions: λ : ( X0 , X1 ..., Xn ) → (λw0 X0 , λw1 X1 , ..., λwn Xn ). (2.5)With this λ we can deﬁne Pnw0 ,...,wn ) = (Cn+1 {0})/C∗ . ( (2.6)Note, that the action of C∗ is no longer free2 . The weighted projective space will thus containquotient singularities. Projective spaces are obviously the most simple examples of toric varieties. The fans (seeSec. 2.1) of P1 and P2 are shown in Figure 2.1. (a) Fan of P1 (b) Fan of P2 Figure 2.1: Fans of projective spaces 2 Suppose wi = 0. Then it is possible to choose λ = 1 such that λwi = 1, which results in (0, .., 0, Xi , 0, ..., 0) =(0, ..., 0, λwi X i , 0, ...0). 15
- 17. A lattice and a fanA toric variety XΣ can be encoded by a lattice N which is isomorphic to Zn and its fan Σ.The fan is a collection of strongly convex rational cones in N ⊗Z R with the property thateach face of a cone in Σ is also a cone in Σ and the intersection of two cones in Σ is a face ofeach. The d–dimensional cones in Σ are in one–to–one correspondence with the codimensiond–submanifolds of XΣ . The one–dimensional cones in particular correspond to the divisors inXΣ . The fan Σ can be encoded by the generators of its edges or one–dimensional cones, i.e. byvectors vi ∈ N. To each vi we associate a homogeneous coordinate zi of XΣ . To each of the vicorresponds the divisor Di which is determined by the equation zi = 0. The (C∗ )m action onthe vi is encoded in m linear relations d ∑ li ( a) ( a) vi = 0, a = 1, . . . , m, li ∈ Z. (2.7) i =1To each linear relation we assign a monomial d ( a) li Ua = ∏ zi . (2.8) i =1These monomials are invariant under the scaling action and form the local coordinates of Xσ . a aIn general, monomials of type z11 ....zkk are sections of line bundles O( a1 D1 + ... + ak Dk ). LetM be the lattice dual to N with respect to the pairing , . For any p ∈ M, monomials of v ,p v ,pthe form z1 1 ....zk k are invariant under the scaling action and thus give rise to a linearequivalence relation v1 , p D1 + ... + vk , p Dk ∼ 0 . (2.9) We are uniquely interested in Calabi–Yau manifolds, therefore we require XΣ to havetrivial canonical class. The canonical divisor of XΣ is given by − D1 − ... − Dn , so for XΣ tobe Calabi–Yau, D1 + ... + Dn must be trivial, i.e. there must be a p ∈ M such that vi , p = 1for every i. This translates to requiring that the vi must all lie in the same afﬁne hyperplaneone unit away from the origin v0 . In our 3–dimensional case, we can choose e.g. the thirdcomponent of all the vectors vi (except v0 ) to equal one. The vi form a cone C (∆(2) ) overthe triangle ∆(2) = v1 , v2 , v3 with apex v0 . The Calabi–Yau condition therefore allows us todraw toric diagrams ∆(2) in two dimensions only. The toric diagram drawn on the hyperplanehas an obvious SL(2, Z) symmetry, i.e. toric diagrams which are connected by an SL(2, Z)transformation give rise to the same toric variety. In the dual diagram, the geometry and intersection properties of a toric manifold areoften easier to grasp than in the original toric diagram. The divisors, which are representedby vertices in the original toric diagram become faces in the dual diagram, the curves markingthe intersections of two divisors remain curves and the intersections of three divisors whichare represented by the faces of the original diagram become vertices. In the dual graph, it isimmediately clear, which of the divisors and curves are compact. For now, we remain with the orbifold examples discussed earlier. So how do we go aboutﬁnding the fan of a speciﬁc C3 /Zn –orbifold? We have just one three–dimensional cone in Σ,generated by v1 , v2 , v3 . The orbifold acts as follows on the coordinates of C3 : θ : ( z 1 , z 2 , z 2 ) → ( ε z 1 , ε n1 z 2 , ε n2 z 3 ), ε = e2πi/n . (2.10) 16
- 18. 1For such an action we will use the shorthand notation n (1, n1 , n2 ). The coordinates of XΣ aregiven by (v ) (v ) (v ) U i = z1 1 i z2 2 i z3 3 i . (2.11)To ﬁnd the coordinates of the generators vi of the fan, we require the U i to be invariant underthe action of θ. We end up looking for two linearly independent solutions of the equation ( v1 )i + n1 ( v2 )i + n2 ( v3 )i = 0 mod n. (2.12)The Calabi–Yau condition is trivially fulﬁlled since the orbifold actions are chosen such that1 + n1 + n2 = n and εn = 1. XΣ is smooth if all the top–dimensional cones in Σ have volume one. By computing thedeterminant det(v1 , v2 , v3 ), it can be easily checked that this is not the case in any of ourorbifolds. We will therefore resolve the singularities by blowing them up.Example A.1: C3 /Z6− IThe group Z6− I acts as follows on C3 : θ : ( z1 , z2 , z3 ) → ( ε z1 , ε z2 , ε4 z3 ), ε = e2πi/6 . (2.13)To ﬁnd the components of the vi , we have to solve ( v1 )i + ( v2 )i + 4 ( v3 )i = 0 mod 6 . (2.14)This leads to the following three generators of the fan (or some other linear combination thereof): 1 −1 0 v1 = −2 , v2 = −2 , v3 = 1 . (2.15) 1 1 1The toric diagram of C3 /Z6− I and its dual diagram are depicted in Figure 2.2. D3 D3 D2 D1 D2 D1 Figure 2.2: Toric diagram of C3 /Z6− I and dual graph 17
- 19. 2.2 Resolution of singularitiesThere are several ways of resolving a singularity, one of them being the blow–up. The processof blowing up consists of two steps in toric geometry: First, we must reﬁne the fan, then sub-divide it. Reﬁning the fan means adding 1–dimensional cones. The subdivision correspondsto choosing a triangulation for the toric diagram. Together, this corresponds to replacing thepoint that is blown up by an exceptional divisor. We denote the reﬁned fan by Σ. ˜ We are interested in resolving the orbifold–singularities such that the canonical class ofthe manifold is not affected, i.e. the resulting manifold is still Calabi–Yau (in mathematicsliterature, this is called a crepant resolution). When adding points that lie in the intersectionof the simplex with corners vi and the lattice N, the Calabi–Yau criterion is met. Aspinwallstudies the resolution of singularities of type Cd /G and gives a very simple prescription [1].We ﬁrst write it down for the case of C3 /Zn . For what follows, it is more convenient to writethe orbifold twists in the form θ : (z1 , z2 , z3 ) → (e2πig1 z1 , e2πig2 z2 , e2πig3 z3 ). (2.16)The new generators wi are obtained via (i ) (i ) (i ) w i = g1 v 1 + g2 v 2 + g3 v 3 , (2.17) (i ) (i ) (i )where the g(i) = ( g1 , g2 , g3 ) ∈ Zn = {1, θ, θ 2 , ... , θ n−1 } such that 3 ∑ gi = 1, 0 ≤ gi < 1. (2.18) i =1θ always fulﬁlls this criterion. We denote the the exceptional divisors corresponding to the wiby Ei . To each of the new generators we associate a new coordinate which we denote by yi ,as opposed to the zi we associated to the original vi . Let us pause for a moment to think about what this method of resolution means. Theobvious reason for enforcing the criterion (2.18) is that group elements which do not respectit fail to fulﬁll the Calabi–Yau condition: Their third component is no longer equal to one. Butwhat is the interpretation of these group elements that do not contribute? Another way tophrase the question is: Why do not all twisted sectors contribute exceptional divisors? A closer 1look at the group elements shows that all those elements of the form n (1, n1 , n2 ) which fulﬁll 1(2.18) give rise to inner points of the toric diagram. Those of the form n (1, 0, n − 1) lead topoints on the edge of the diagram. They always fulﬁll (2.18) and each element which belongsto such a sub–group contributes a divisor to the respective edge, therefore there will be n − 1points on it. The elements which do not fulﬁll (2.18) are in fact anti–twists, i.e. they have the 1form n (n − 1, n − n1 , n − n2 ). Since the anti–twist does not carry any information which wasnot contained already in the twist, there is no need to take it into account separately, so alsofrom this point of view it makes sense that it does not contribute an exceptional divisor to theresolution. The case C2 /Zn is even simpler. The singularity C2 /Zn is called a rational double pointof type An−1 and its resolution is called a Hirzebruch–Jung sphere tree consisting of n − 1exceptional divisors intersecting themselves according to the Dynkin diagram of An−1 . Thecorresponding polyhedron ∆(1) consists of a single edge joining two vertices v1 and v2 withn − 1 equally spaced lattice points w1 , . . . , wn−1 in the interior of the edge. 18
- 20. Now we subdivide the cone. The diagram of the resolution of C3 /G contains n triangles,where n is the order of G, yielding n three–dimensional cones. For most groups G, severaltriangulations, and therefore several resolutions are possible (for large group orders even sev-eral thousands). They are all related via birational3 transformations, namely ﬂop transitions.Some physical properties change for different triangulations, such as the intersection ring.Different triangulations correspond to different phases in the Kähler moduli space. This treatment is easily extended to C3 /Z N × Z M –orbifolds. When constructing the fan,the coordinates of the generators vi not only have to fulﬁll one equation (2.12) but three,coming from the twist θ 1 associated to Z N , the twist θ 2 associated to Z M and from thecombined twist θ 1 θ 2 . When blowing up the orbifold, the possible group elements g(i) are {(θ 1 )i (θ 2 ) j , i = 0, ..., N − 1, j = 0, ..., M − 1}. (2.19)The toric diagram of the blown–up geometry contains N · M triangles corresponding to thetree–dimensional cones. The remainder of the preceding discussion remains the same. We also want to settle the question to which toric variety the blown–up geometry corre-sponds. Applied to our case XΣ = C3 /G, the new blown up variety corresponds to XΣ = C3+d FΣ /(C∗ )d , ˜ ˜ (2.20)where d is the number of new generators wi of one–dimensional cones. The action of (C∗ )dcorresponds to the set of rescalings that leave the (v ) (v ) (v ) Ui = z1 1 i z2 2 i z3 3 i (y1 )(w1 )i... (yd )(wd )i ˜ (2.21)invariant. The excluded set FΣ is determined as follows: Take the set of all combinations ˜of generators vi of one–dimensional cones in Σ that do not span a cone in Σ and deﬁne foreach such combination a linear space by setting the coordinates associated to the vi to zero.FΣ is the union of these linear spaces, i.e. the set of simultaneous zeros of coordinates notbelonging to the same cone. In the case of several possible triangulations, it is the excludedset that distinguishes the different resulting geometries.Example A.1: C3 /Z6− IWe will now resolve the singularity of C3 /Z6− I . The group elements are θ = 1 (1, 1, 4), θ 2 = 613 (1, 1, 1), θ 3 = 1 (1, 1, 0), θ 4 = 3 (2, 2, 2) and θ 5 = 6 (5, 5, 2). θ, θ 2 and θ 3 fulﬁll condition 2 1 1(2.18). This leads to the following new generators: 1 w1 = 6 v1 + 1 v2 + 4 v3 = (0, 0, 1), 6 6 1 w2 = 3 v1 + 1 v2 + 1 v3 = (0, −1, 1), 3 3 1 w3 = 2 v1 + 1 v2 = (0, −2, 1). 2 (2.22)In this case, the triangulation is unique. Figure 2.3 shows the corresponding toric diagram and ˜its dual graph. Let us identify the new geometry. The Ui are 3 A birational map between algebraic varieties is a rational map with a rational inverse. A rational map from acomplex manifold M to projective space Pn is a map f : z → [1, f 1 (z), ..., f n (z)] given by n global meromorphicfunctions on M. 19
- 21. D3 D3 E1 D2 D1 E2 E1 E2 E3 D2 E3 D1 Figure 2.3: Toric diagram of the resolution of C3 /Z6− I and dual graph ˜ z1 ˜ z3 ˜ U1 = , U2 = 2 z2 y y2 , U3 = z1 z2 z3 y1 y2 y3 . (2.23) z2 z1 2 2 3 ˜The rescalings that leave the Ui invariant are 1 ( z1 , z2 , z3 , y1 , y2 , y3 ) → ( λ1 z1 , λ1 z2 , λ4 λ2 λ3 z3 , 1 y1 , λ2 y2 , λ3 y3 ). (2.24) λ6 λ2 λ3 1 2 3According to eq. (2.20), the new blown–up geometry is XΣ = (C6 FΣ )/(C∗ )3 , ˜ ˜ (2.25)where the action of (C∗ )3 is given by eq. (2.24). The excluded set is generated by FΣ = {(z3 , y2 ) = 0, (z3 , y3 ) = 0, (y1 , y3 ) = 0, (z1 , z2 ) = 0 }. ˜ As can readily be seen in the dual graph, we have seven compact curves in XΣ . Two of them, ˜{y1 = y2 = 0} and {y2 = y3 = 0} are exceptional. They both have the topology of P1 . Take forexample C1 : To avoid being on the excluded set, we must have y3 = 0, z3 = 0 and (z1 , z2 ) = 0.Therefore C1 = {(z1 , z2 , 1, 0, 0, 1), (z1 , z1 ) = 0}/(z1 , z2 ), which corresponds to a P1 . We have now six three–dimensional cones: S1 = ( D1 , E2 , E3 ), S2 = ( D1 , E2 , E1 ), S3 =( D1 , E1 , D3 ), S4 = ( D2 , E2 , E3 ), S5 = ( D2 , E2 , E1 ), and S6 = ( D2 , E1 , D3 ).Example B.1: C3 /Z6− I IWe brieﬂy give another example to illustrate the relation between different triangulations of atoric diagram. The resolution of C3 /Z6− I I allows ﬁve different triangulations. Figure 2.4 givesthe ﬁve toric diagrams. We start out with triangulation a). When the curve D1 · E1 is blown down and the curveE3 · E4 is blown up instead, we have gone through a ﬂop transition and arrive at the triangulationb). From b) to c) we arrive by performing the ﬂop E1 · E4 → E2 · E3 . From c) to d) takes us theﬂop E1 · E2 → D2 · E3 . The last triangulation e) is produced from b) by ﬂopping E1 · E3 → D3 · E4 .Thus, all triangulations are related to each other by a series of birational transformations. 20
- 22. a) b) c) D3 D3 D3 E3 E1 E3 E1 E3 E1 D1 E4 E2 D2 D1 E4 E2 D2 D1 E4 E2 D2 d) e) D3 D3 E3 E1 E3 E1 D1 E4 E2 D2 D1 E4 E2 D2Figure 2.4: The ﬁve different triangulations of the toric diagram of the resolution of C3 /Z6− I I2.3 Mori cone and intersection numbersThe intersection ring of a variety is an important quantity which often proves to be of interestfor the physicist (e.g. to determine the Kähler potential for the Kähler moduli space). Theframework of toric geometry allows us to extract the desired information with ease. Note that the intersection number of two cycles A, B only depends on the homologyclasses of A and B. Note also that ∑ bi Di and ∑ bi Di (where the Di are the divisors corre-sponding to the one–dimensional cones) are linearly equivalent if and only if they are homo-logically equivalent. To arrive at the equivalences in homology, we ﬁrst identify the linear relations betweenthe divisors of the form i i i i i a1 v1 + a2 v2 + a3 v3 + a4 w1 + ... + a3+d wd = 0 . (2.26)These linear relations can be obtained either by direct examination of the generators or can beread off directly from the algebraic torus action (C∗ )m . The exponents of the different scalingparameters yield the coefﬁcients ai . The divisors corresponding to such a linear combinationare sliding divisors in the compact geometry. It is very convenient to introduce a matrix( P | Q ): The rows of P contain the coordinates of the vectors vi and wi . The columns ofQ contain the linear relations between the divisors, i.e. the vectors { ai }. From the rows ofQ, which we denote by Ci , i = 1, ..., d, we can read off the linear equivalences in homologybetween the divisors which enable us to compute all triple intersection numbers. For mostapplications, it is most convenient to choose the Ci to be the generators of the Mori cone. TheMori cone is the space of effective curves, i.e. the space of all curves C ∈ XΣ with C · D ≥ 0 for all divisors D ∈ XΣ . (2.27)It is dual to the Kähler cone. In our cases, the Mori cone is spanned by curves correspondingto two–dimensional cones. The curves correspond to the linear relations for the vertices. The 21
- 23. generators for the Mori cone correspond to those linear relations in terms of which all otherscan be expressed as positive, integer linear combinations. We will brieﬂy survey the method of ﬁnding the generators of the Mori cone, which canbe found e.g. in [2]. I. In a given triangulation, take the three–dimensional simplices Sk (corresponding to the three–dimensional cones). Take those pairs of simplices (Sl , Sk ) that share a two– dimensional simplex Sk ∩ Sl . II. For each such pair ﬁnd the unique linear relation among the vertices in Sk ∪ Sl such that (i) the coefﬁcients are minimal integers and (ii) the coefﬁcients for the points in (Sk ∪ Sl ) (Sk ∩ Sl ) are positive. III. Find the minimal integer relations among those obtained in step 2 such that each of them can be expressed as a positive integer linear combination of them.While the ﬁrst two steps are very simple, step III. becomes increasingly tricky for largergroups. The general rule for triple intersections is that the intersection number of three distinctdivisors is 1 if they belong to the same cone and 0 otherwise. The set of collections of divisorswhich do not intersect because they do not lie in the same come forms a further characteristicquantity of a toric variety, the Stanley–Reisner ideal. It contains the same information as theexceptional set FΣ . Intersection numbers for triple intersections of the form Di2 D j or Ek can 3be obtained by making use of the linear equivalences between the divisors. Since we areworking here with non–compact varieties at least one compact divisor has to be involved. Forintersections in compact varieties there is no such condition. The intersection ring of a toricvariety is – up to a global normalization – completely determined by the linear relations andthe Stanley–Reisner ideal. The normalization is ﬁxed by one intersection number of threedistinct divisors. The matrix elements of Q are the intersection numbers between the curves Ci and the divi-sors Di , Ei . We can use this to determine how the compact curves of our blown–up geometryare related to the Ci .Example A.1: C3 /Z6− IFor this example, the method of working out the Mori generators is shown step by step. We givethe pairs, the sets Sl ∪ Sk (the points underlined are those who have to have positive coefﬁcients)and the linear relations: 1. S6 ∪ S3 = { D1 , D2 , D3 , E1 }, D1 + D2 + 4 D3 − 6 E1 = 0, 2. S5 ∪ S2 = { D1 , D2 , E1 , E2 }, D1 + D2 + 2 E1 − 4 E2 = 0, 3. S4 ∪ S1 = { D1 , D2 , E2 , E3 }, D1 + D2 − 2 E3 = 0, 4. S3 ∪ S2 = { D1 , D3 , E1 , E2 }, D3 − 2 E1 + E2 = 0, 5. S2 ∪ S1 = { D1 , E1 , E2 , E3 }, E1 − 2 E2 + E3 = 0, 6. S6 ∪ S5 = { D2 , D3 , E1 , E2 }, D3 − 2 E1 + E2 = 0, 7. S5 ∪ S4 = { D2 , E1 , E2 , E3 }, E1 − 2 E2 + E3 = 0. (2.28) 22
- 24. Curve D1 D2 D3 E1 E2 E3 E1 · E2 1 1 0 2 -4 0 E2 · E3 1 1 0 0 0 -2 D1 · E1 0 0 1 -2 1 0 D1 · E2 0 0 0 1 -2 1 D2 · E1 0 0 1 -2 1 0 D2 · E2 0 0 0 1 -2 1 D3 · E1 1 1 4 -6 0 0 Table 2.1: Triple intersection numbers of the blow–up of Z6− I IWith the relations 3, 4 and 5 all other relations can be expressed as a positive integer linearcombination. This leads to the following three Mori generators: C1 = {0, 0, 0, 1, −2, 1}, C2 = {1, 1, 0, 0, 0, −2}, C3 = {0, 0, 1, −2, 1, 0}. (2.29)With this, we are ready to write down ( P | Q): D1 1 −2 | 0 1 0 1 D2 −1 −2 1 | 0 1 0 D 0 1 1 | 0 0 1 ( P | Q) = 3 . (2.30) E1 0 0 1 | 1 0 −2 E2 0 −1 1 | −2 0 1 E3 0 −2 1 | 1 −2 0From the rows of Q, we can read off directly the linear equivalences: D1 ∼ D2 , E2 ∼ −2 E1 − 3 D3 , E3 ∼ E1 − 2 D1 + 2 D3 . (2.31)The matrix elements of Q contain the intersection numbers of the Ci with the D1 , E1 , e.g. E1 ·C3 = −2, D3 · C1 = 0, etc. We know that E1 · E3 = 0. From the linear equivalences between thedivisors, we ﬁnd the following relations between the curves Ci and the seven compact curves ofour geometry: C1 = D1 · E2 = D2 · E2 , (2.32a) C2 = E2 · E3 , (2.32b) C3 = D1 · E1 = D2 · E1 , (2.32c) E1 · E2 = 2 C1 + C2 , (2.32d) D3 · E1 = 2 C1 + C2 + 4 C3 . (2.32e)From these relations and ( P | Q), we can get all triple intersection numbers, e.g. 2 E1 E2 = E1 E2 E3 + 2 D1 E1 E2 − 2 D3 E1 E2 = 2 . (2.33)Table 2.1 gives the intersections of all compact curves with the divisors. Using the linear equivalences, we can also ﬁnd the triple self–intersections of the compactexceptional divisors: 3 3 E1 = E2 = 8 . (2.34)From the intersection numbers in Q, we ﬁnd that { E1 + 2 D3 , D2 , D3 } form a basis of the Kählercone which is dual to the basis {C1 , C2 , C3 } of the Mori cone. 23
- 25. (a) fan of the Hirzebruch surface (b) fan of dP0 = P2 (c) fan of P1 × P1 = F0 Fn (d) fan of dP1 = Bl1 P2 (e) fan of dP2 = Bl2 P2 (f) fan of dP3 = Bl3 P2 Figure 2.5: Fans of Fn and the toric del Pezzo surfaces2.4 Divisor topologiesThere are two types of exceptional divisors: The compact divisors, whose correspondingpoints lie in the interior of the toric diagram, and the semi–compact ones whose points siton the boundary of the toric diagram. The latter case corresponds to the two–dimensionalsituation with an extra non–compact direction, hence it has the topology of C × P1 withpossibly some blow–ups. The D–divisors are non–compact and of the form C2 . We ﬁrst discuss the compact divisors. For this purpose we use the notion of the star ofa cone σ, in terms of which the topology of the corresponding divisor is determined. Thestar, denoted Star(σ) is the set of all cones τ in the fan Σ containing σ. This means that wesimply remove from the fan Σ all cones, i.e. points and lines in the toric diagram, which donot contain wi . The diagram of the star is not necessarily convex anymore. Then we computethe linear relations and the Mori cone for the star. This means in particular that we drop allthe simplices Sk in the induced triangulation of the star which do not lie in its toric diagram.As a consequence, certain linear relations of the full diagram will be removed in the processof determining the Mori cone. The generators of the Mori cone of the star will in general bedifferent from those of Σ. Once we have obtained the Mori cone of the star, we can rely onthe classiﬁcation of compact toric surfaces. 24
- 26. Digression: Classiﬁcation of compact toric surfaces Any toric surface is either a P2 , a Hirzebruch surface Fn , or a toric blow–up thereof. The simplest possible surface is obviously P2 . Each surface, which is birationally equivalent to P2 is called a rational surface. Hirzebruch surfaces A Hirzebrucha surface Fn is a special case of a ruled surface S, which admits a ﬁbration π : S → C, C a smooth curve and the generic ﬁber of π being isomorphic to P1 . A Hirzebruch surface is a ﬁbration of P1 over P1 and is of the form Fn = P(OP1 ⊕ OP1 (n)). del Pezzo surfaces A del Pezzob surface is a two–dimensional Fano variety, i.e. a variety whose anticanonical bundle is amplec . In total, there exist 10 of them: dP0 = P2 , P1 × P1 = F0 and blow–ups of P2 in up to 8 points, Bln P2 = dPn . Five of them are realized as toric surfaces, namely F0 and dPn , n = 0, ..., 3. The fans are given in Figure 2.5. In Figure 2.5.a, the fan of Fn is shown. a FriedrichE.P. Hirzebruch (*1927), German mathematician b Pasquale del Pezzo (1859-1936), Neapolitan mathematician c A line bundle L is very ample, if it has enough sections to embed its base manifold into projective space. L is ample, if a tensor power L⊗n of L is very ample. The generator of the Mori cone of P2 has the form QT = −3 1 1 1 . (2.35)For Fn , the generators take the form −2 1 1 0 0 −2 1 1 0 0 QT = or QT = (2.36) −n − 2 0 n 1 1 n − 2 0 −n 1 1since F−n is isomorphic to Fn . Finally, every toric blow–up of a point adds an additionalindependent relation whose form is QT = 0 ... 0 1 1 −2 . (2.37)We will denote the blow–up of a surface S in n points by Bln S. The toric variety XΣ isthree dimensional, which means in particular that the stars are in fact cones over a polygon. 25
- 27. An additional possibility for a toric blow–up is adding a point to the polygon such that thecorresponding relation is of the form QT = 0 ... 0 1 1 −1 −1 . (2.38)This corresponds to adding a cone over a lozenge and is well–known from the resolution ofthe conifold singularity. Also the semi–compact exceptional divisors can be dealt with using the star. Since thegeometry is effectively reduced by one dimension, the only compact toric manifold in onedimension is P1 and the corresponding generator is QT = −2 1 1 0 , (2.39)where the 0 corresponds to the non–compact factor C.Example A.1: C3 /Z6− IWe now determine the topology of the exceptional divisors for our example C3 /Z6− I . As ex- D3 E1 E1 E2 E2 E2 D2 D1 D2 E3 D1 D2 E3 D1 Figure 2.6: The stars of the exceptional divisors E1 , E2 , and E3 , respectively.plained above, we need to look at the respective stars which are displayed in Figure 2.6. In orderto determine the Mori generators for the star of E1 , we have to drop the cones involving E3 whichare S1 and S4 . From the seven relations in (2.28) only four remain, those corresponding to C3 ,2 C1 + C2 and 2 C1 + C2 + 4 C3 . These are generated by 2 C1 + C2 = (1, 1, 0, 2, −4, 0) and (2.40) C3 = (0, 0, 1, −2, 1, 0), (2.41)which are the Mori generators of F4 . Similarly, for the star of E2 only the relations not involvingS3 and S6 remain. These are generated by C1 and C2 , and using (2.29) we recognize them tobe the Mori generators of F2 . Finally, the star of E3 has only the relation corresponding to C3 .Hence, the topology of E3 is P1 × C, as it should be, since the point sits on the boundary of thetoric diagram of XΣ and no extra exceptional curves end on it. 26
- 28. 2.5 LiteratureFor a ﬁrst acquaintance with toric geometry, Chapter 7 of [3] is well suited. Also [4] containsa very readable introduction. The classical references on toric geometry are the books byFulton [5] and Oda [6]. Unfortunately, they are both not very accessible to the physicist. Thereference for general techniques in algebraic geometry is [7]. A number of reviews of topological string theory brieﬂy introduce toric geometry, such as[8, 9], but from a different point of view.Bibliography[1] P. S. Aspinwall, Resolution of orbifold singularities in string theory, hep-th/9403123.[2] P. Berglund, S. H. Katz, and A. Klemm, Mirror symmetry and the moduli space for generic hypersurfaces in toric varieties, Nucl. Phys. B456 (1995) 153–204, [hep-th/9506091].[3] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry. No. 1 in Clay Mathematics Monographs. American Mathematical Society, Clay Mathematics Institute, 2003.[4] P. S. Aspinwall, B. R. Greene, and D. R. Morrison, Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nucl. Phys. B416 (1994) 414–480, [hep-th/9309097].[5] W. Fulton, Introduction to Toric Varieties. Princeton University Press, Princeton, New Jersey, 1993.[6] T. Oda, Lectures on Torus Embeddings and Applications. Tata Institute of Fundamental Research, Narosa Publishing House, New Delhi, 1978.[7] P. Grifﬁths and J. Harris, Principles of Algebraic Geometry. John Wiley and Sons, Inc., 1978.[8] A. Neitzke and C. Vafa, Topological strings and their physical applications, hep-th/0410178.[9] M. Marino, Chern-Simons theory and topological strings, Rev. Mod. Phys. 77 (2005) 675–720, [hep-th/0406005]. 27
- 29. Lecture 3Application: Desingularizing toroidalorbifoldsContents 3.1 Gluing the patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 The inherited divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 The intersection ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Divisor topologies for the compact manifold . . . . . . . . . . . . . . . . . 39 In this lecture, I will discuss the desingularization of toroidal orbifolds employing themethods treated so far. First, I explain how to glue together the resolved toric patches toobtain a smooth Calabi–Yau manifold from the singular orbifold quotient T 6 /Γ. Next, thedivisors inherited directly from the covering space T 6 are discussed. In the following section,the full intersection ring of the smooth manifold is calculated, and lastly, the topologies of theappearing divisor classes are determined.3.1 Gluing the patchesIn the easy cases, say in the prime orbifolds Z3 and Z7 , it is obvious how the smooth manifoldis obtained: Just put one resolved patch in the location of every ﬁxed point and you areﬁnished. Since these patches only have internal points, the corresponding exceptional divisorsare compact, hence cannot see each other, and no complications arise from gluing. Fixed lines which do not intersect any other ﬁxed lines and on top of which no ﬁxed pointssit also pose no problem. But what happens, when we have ﬁxed lines on top of which ﬁxed points are sitting? Asdiscussed in Section 2.2, such a ﬁxed point already knows it sits on a ﬁxed line, since on theedge of the toric diagram of its resolution is the number of exceptional divisors appropriate tothe ﬁxed line the point sits on top of. Internal exceptional divisors are unproblematic in thiscase as well, since they do not feel the global surrounding. The exceptional divisors on theedges are identiﬁed or glued together with those of the corresponding resolved ﬁxed lines. The larger the order of the group, the more often it happens that a point or line is ﬁxedunder several group elements. How are we to know which of the patches we should use? 28
- 30. In the case of ﬁxed lines answer is: use the patch that belongs to the generator of thelargest subgroup under which the patch is ﬁxed, because the line is ﬁxed under the wholesub-group and its exceptional divisors already count the contributions from the other groupelements. For ﬁxed points, the question is a little more tricky. One possibility is to count thenumber of group elements this point is ﬁxed under, not counting anti–twists and elementsthat generate ﬁxed lines. Then choose the patch with the matching number of interior points.The other possibility is to rely on the schematic picture of the ﬁxed set conﬁguration andchoose the patch according to the ﬁxed lines the ﬁxed point sits on. Isolated ﬁxed pointscorrespond to toric diagrams with only internal, compact exceptional divisors. When theﬁxed point sits on a ﬁxed line of order k, its toric diagram has k − 1 exceptional divisors onone of its boundaries. If the ﬁxed point sits at the intersection of two (three) ﬁxed lines, it hasthe appropriate number of exceptional divisors on two (three) of its boundaries. The rightnumber of interior points together with the right number of exceptional divisors sitting onthe edges uniquely determines the correct patch. Even though the intersection points of three Z2 ﬁxed lines are not ﬁxed under a singlegroup element, they must be resolved. The resolution of such a point is the resolution ofC3 /Z2 × Z2 and its toric diagram is indeed the only one without interior points, see Fig-ure 3.1. D1 E3 E2 D2 E1 D3 Figure 3.1: Toric diagram of resolution of C3 /Z2 × Z2 and dual graph Interestingly, the case of three intersecting Z2 ﬁxed lines is the only instance of intersect-ing ﬁxed lines where the intersection point itself is not ﬁxed under a single group element.This case arises only for Zn × Zm orbifolds with both n and m even.Example A: Z6− I on G2 × SU (3) 2This example is rather straightforward. We must again use the data of Table 1.5 and theschematic picture of the ﬁxed set conﬁguration 1.2. Furthermore, we need the resolved patches ofC3 /Z6− I (see Section 2.2, in particular Figure 2.3), C3 /Z3 (see Figure 3.2), and the resolutionof the Z2 ﬁxed line. The three Z6 –patches contribute two exceptional divisors each: E1,γ , andE2,1,γ , where γ = 1, 2, 3 labels the patches in the z3 –direction. The exceptional divisor E3 on theedge is identiﬁed with the one of the resolved ﬁxed line the patch sits upon, as we will see. Thereare furthermore 15 conjugacy classes of Z3 ﬁxed points. Blowing them up leads to a contributionof one exceptional divisor as can be seen from Figure 3.2. Since three of these ﬁxed points sitat the location of the Z6− I ﬁxed points which we have already taken into account (E2,1,γ ), we 29
- 31. D3 E D2 D1 Figure 3.2: Toric diagram of the resolution of C3 /Z3only count 12 of them, and denote the resulting divisors by E2,µ,γ , µ = 2, . . . , 5, γ = 1, 2, 3. Theinvariant divisors are built according to the conjugacy classes, e.g. E2,2,γ = E2,1,2,γ + E2,1,3,γ , (3.1)etc., where E2,α,β,γ are the representatives on the cover. Finally, there are 6 conjugacy classesof ﬁxed lines of the form C2 /Z2 . We see that after the resolution, each class contributes one ﬁxed,1 = zﬁxed,1 = 0 sit the three Z6− Iexceptional divisor E3,α , α = 1, 2. On the ﬁxed line at z1 2ﬁxed points. The divisor coming from the blow–up of this ﬁxed line, E3,1 , is identiﬁed with thethree exceptional divisors corresponding to the points on the boundary of the toric diagram of theresolution of C3 /Z6− I that we mentioned above. In total, this adds up to h1,1 twisted = 3 · 2 + 12 · 1 + 6 · 1 = 24 (3.2) (1,1)exceptional divisors, which is the number which is given for htwisted in Table 1.3.Example C: T 6 /Z6 × Z6This, being the point group of largest order, is the most tedious of all examples. It is presentedhere to show that the procedure is not so tedious after all. First, the ﬁxed sets must be identiﬁed. Table 3.1 summarizes the results. Figure 3.3 showsthe schematic picture of the ﬁxed set conﬁguration. Again, it is the covering space that is shown,the representatives of the equivalence classes are highlighted. Now we are ready to glue the patches together. Figure 3.4 schematically shows all the patchesthat will be needed in this example. It is easiest to ﬁrst look at the ﬁxed lines. There are three Z6ﬁxed lines, each contributing ﬁve exceptional divisors. Then there are twelve equivalence classesof Z3 ﬁxed lines, three of which coincide with the Z6 ﬁxed lines. The latter need not be counted,since they are already contained in the divisor count of the Z6 ﬁxed lines. The Z3 ﬁxed lineseach contribute two exceptional divisors. Furthermore, there are twelve equivalence classes of Z2ﬁxed lines, three of which again coincide with the Z6 ﬁxed lines. They give rise to one exceptionaldivisor each. From the ﬁxed lines originate in total h1,1 = 3 · 5 + (12 − 3) · 2 + (12 − 3) · 1 = 42 lines (3.3) 30
- 32. 1 25 Z3 1 2 24 Z6 θ (θ ) (θ ) (θ ) z3 z3 2 2 2 1 θ 1 2 (θ ) θ (θ ) z1 z1 z2 z2 Z2 1 3 23 1 2 (θ ) (θ ) z3 θθ z3 2 3 1 3 (θ )(θ ) z1 z2 z1 z2 1 24 14 2 θ (θ ) (θ ) θ 1 2 22 z3 (θ ) (θ ) z3 z2 z1 z1 z2 1 22 1 2 2 θ (θ ) (θ ) θ z3 z3 z1 z1 z2 z2 1 3 2 2 1 3 2 1 2 2 3 (θ ) (θ ) 1 23 (θ ) θ (θ ) (θ ) θ (θ ) Figure 3.3: Schematic picture of the ﬁxed set conﬁguration of Z6 × Z6 31
- 33. Group el. Order Fixed Set Conj. Classes θ1 6 1 ﬁxed line 1 ( θ 1 )2 3 9 ﬁxed lines 4 ( θ 1 )3 2 16 ﬁxed lines 4 θ2 6 1 ﬁxed line 1 ( θ 2 )2 3 9 ﬁxed lines 4 ( θ 2 )3 2 16 ﬁxed lines 4 θ1 θ2 6×6 3 ﬁxed points 2 θ 1 ( θ 2 )2 6×3 12 ﬁxed points 4 θ 1 ( θ 2 )3 6×2 12 ﬁxed points 4 θ 1 ( θ 2 )4 6×6 3 ﬁxed points 2 θ 1 ( θ 2 )5 6 1 ﬁxed line 1 ( θ 1 )2 θ 2 3×6 12 ﬁxed points 4 ( θ 1 )3 θ 2 2×6 12 ﬁxed points 4 ( θ 1 )4 θ 2 6×6 3 ﬁxed points 2 ( θ 1 )2 ( θ 2 )2 3×3 27 ﬁxed points 9 ( θ 1 )2 ( θ 2 )3 3×2 12 ﬁxed points 4 ( θ 1 )2 ( θ 2 )4 3 9 ﬁxed lines 4 ( θ 1 )3 ( θ 2 )2 2×3 12 ﬁxed points 4 ( θ 1 )3 ( θ 2 )3 2 16 ﬁxed lines 4 Table 3.1: Fixed point set for Z6 × Z6 . 3 3 C /Z2 x Z2 C /Z2 x Z3 3 3 C /Z2 x Z6 C /Z3 x Z3 3 3 C /Z3 x Z6 C /Z6 x Z6 Figure 3.4: Toric diagrams of patches for T 6 /Z6 × Z6exceptional divisors. Now we study the ﬁxed points. We associate the patches to the ﬁxed points according to the 32
- 34. intersection of ﬁxed lines on which they sit. The exceptional divisors on the boundaries of theirtoric diagrams are identiﬁed with the divisors of the respective ﬁxed lines. There is but one ﬁxedpoint on the intersection of three Z6 ﬁxed lines. It is replaced by the resolution of the C2 /Z6 × Z6patch, which contributes ten compact internal exceptional divisors. There are three equivalenceclasses of ﬁxed points on the intersections of one Z6 ﬁxed line and two Z3 ﬁxed lines. They arereplaced by the resolutions of the C2 /Z3 × Z6 patch, which contribute four compact exceptionaldivisors each. Then, there are ﬁve equivalence classes of ﬁxed points on the intersections of threeZ3 ﬁxed lines. They are replaced by the resolutions of the C2 /Z3 × Z3 patch, which contributeone compact exceptional divisor each. Furthermore, there are three equivalence classes of ﬁxedpoints on the intersections of one Z6 ﬁxed line and two Z2 ﬁxed lines. They are replaced by theresolutions of the C2 /Z2 × Z6 patch, which contribute two compact exceptional divisors each.The rest of the ﬁxed points sit on the intersections of one Z2 and one Z3 ﬁxed line. There aresix equivalence classes of them. They are replaced by the resolutions of the C2 /Z2 × Z3 patch,which is the same as the C2 /Z6− I I patch, which contribute one compact exceptional divisor each.On the intersections of three Z2 ﬁxed lines sit resolved C2 /Z2 × Z2 patches, but since this patchhas no internal points, it does not contribute any exceptional divisors which were not alreadycounted by the ﬁxed lines. The ﬁxed points therefore yield h1,1 = 1 · 10 + 3 · 4 + 5 · 1 + 3 · 2 + 6 · 1 = 39 pts (3.4)exceptional divisors. From ﬁxed lines and ﬁxed points together we arrive at h1,1 twisted = 42 + 39 = 81 (3.5)exceptional divisors.3.2 The inherited divisorsSo far, we have mainly spoken about the exceptional divisors which arise from the blow–upsof the singularities. In the local patches, the other natural set of divisors are the D–divisors,which descend from the local coordinates zi of the C3 –patch. On the compact space, i.e. the ˜resolution of T 6 /Γ, the Ds are not the natural quantities anymore. The natural quantitiesare the divisors Ri which descend from the covering space T 6 and are dual to the untwisted(1, 1)–forms of the orbifold. The three forms dzi ∧ dzi , i = 1, 2, 3 (3.6)are invariant under all twists. For each pair ni = n j in the twist (1.15), the forms dzi ∧ dz j and dz j ∧ dzi (3.7)are invariant as well. The inherited divisors Ri together with the exceptional divisors Ek,α,β,γ form a basis for thedivisor classes of the resolved orbifold. The D–divisors, which in the local patches are deﬁned by zi = 0 are in the compact ˜manifold deﬁned by Diα = {zi = zﬁxed,α }, i (3.8) 33

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